Basically, i'm afraid you can't. Any directory enquiry call line are under absolute strict instructions to never give the number out. They won't even ring them on your behalf to let them know that someone is trying to contact them urgently. It could lead to hoax and cold callers which is possibly why they were ex-d anyway. Unless you have a mutual connection with these people who might be able to help, it looks like the written word will be the only option.

The only hope you have, and this is definitely a long-shot, is to find someone with an old copy of UK Info and hope they maybe had the number on there before going ex-D. As Sweggie says, they are very strict on this one.

It's not only that they won't ring them on your behalf its that they don't have the number shown - all they see on the screen where the numbers are usually shown is X/D and nothing else - this prevents the operators who are suckers for a hard luck story from being fooled into divulging confidential information. Sweggie is right in saying the only way is to get a mutual friend to help or just to stick to pen and paper.

If you get a private investigator they can on occasions get you the ex-directory number, but remember, because you are ex-directory does not mean that your number is withheld, ex-directory means the name is excluded from the directory ONLY, you have to state that you want the number withheld also

If you want an ex-directory phone number usually a BT engineer will obtain it for you or at least a few years ago this was the case.
Also why if some detective and investigative agencies can find Ex D numbers then why is there no avenue to find them on the net ???
Peter
North London